MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dcomex Structured version   Visualization version   GIF version

Theorem dcomex 10384
Description: The Axiom of Dependent Choice implies Infinity, the way we have stated it. Thus, we have Inf+AC implies DC and DC implies Inf, but AC does not imply Inf. (Contributed by Mario Carneiro, 25-Jan-2013.)
Assertion
Ref Expression
dcomex ω ∈ V

Proof of Theorem dcomex
Dummy variables 𝑡 𝑠 𝑥 𝑓 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1n0 8435 . . . . . . 7 1o ≠ ∅
2 df-br 5107 . . . . . . . 8 ((𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) ↔ ⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩})
3 elsni 4604 . . . . . . . . 9 (⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩} → ⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1o, 1o⟩)
4 fvex 6856 . . . . . . . . . 10 (𝑓𝑛) ∈ V
5 fvex 6856 . . . . . . . . . 10 (𝑓‘suc 𝑛) ∈ V
64, 5opth1 5433 . . . . . . . . 9 (⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1o, 1o⟩ → (𝑓𝑛) = 1o)
73, 6syl 17 . . . . . . . 8 (⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩} → (𝑓𝑛) = 1o)
82, 7sylbi 216 . . . . . . 7 ((𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → (𝑓𝑛) = 1o)
9 tz6.12i 6871 . . . . . . 7 (1o ≠ ∅ → ((𝑓𝑛) = 1o𝑛𝑓1o))
101, 8, 9mpsyl 68 . . . . . 6 ((𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → 𝑛𝑓1o)
11 vex 3450 . . . . . . 7 𝑛 ∈ V
12 1oex 8423 . . . . . . 7 1o ∈ V
1311, 12breldm 5865 . . . . . 6 (𝑛𝑓1o𝑛 ∈ dom 𝑓)
1410, 13syl 17 . . . . 5 ((𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → 𝑛 ∈ dom 𝑓)
1514ralimi 3087 . . . 4 (∀𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
16 dfss3 3933 . . . 4 (ω ⊆ dom 𝑓 ↔ ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
1715, 16sylibr 233 . . 3 (∀𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ω ⊆ dom 𝑓)
18 vex 3450 . . . . 5 𝑓 ∈ V
1918dmex 7849 . . . 4 dom 𝑓 ∈ V
2019ssex 5279 . . 3 (ω ⊆ dom 𝑓 → ω ∈ V)
2117, 20syl 17 . 2 (∀𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ω ∈ V)
22 snex 5389 . . 3 {⟨1o, 1o⟩} ∈ V
2312, 12fvsn 7128 . . . . . . . 8 ({⟨1o, 1o⟩}‘1o) = 1o
2412, 12funsn 6555 . . . . . . . . 9 Fun {⟨1o, 1o⟩}
2512snid 4623 . . . . . . . . . 10 1o ∈ {1o}
2612dmsnop 6169 . . . . . . . . . 10 dom {⟨1o, 1o⟩} = {1o}
2725, 26eleqtrri 2837 . . . . . . . . 9 1o ∈ dom {⟨1o, 1o⟩}
28 funbrfvb 6898 . . . . . . . . 9 ((Fun {⟨1o, 1o⟩} ∧ 1o ∈ dom {⟨1o, 1o⟩}) → (({⟨1o, 1o⟩}‘1o) = 1o ↔ 1o{⟨1o, 1o⟩}1o))
2924, 27, 28mp2an 691 . . . . . . . 8 (({⟨1o, 1o⟩}‘1o) = 1o ↔ 1o{⟨1o, 1o⟩}1o)
3023, 29mpbi 229 . . . . . . 7 1o{⟨1o, 1o⟩}1o
31 breq12 5111 . . . . . . . 8 ((𝑠 = 1o𝑡 = 1o) → (𝑠{⟨1o, 1o⟩}𝑡 ↔ 1o{⟨1o, 1o⟩}1o))
3212, 12, 31spc2ev 3567 . . . . . . 7 (1o{⟨1o, 1o⟩}1o → ∃𝑠𝑡 𝑠{⟨1o, 1o⟩}𝑡)
3330, 32ax-mp 5 . . . . . 6 𝑠𝑡 𝑠{⟨1o, 1o⟩}𝑡
34 breq 5108 . . . . . . 7 (𝑥 = {⟨1o, 1o⟩} → (𝑠𝑥𝑡𝑠{⟨1o, 1o⟩}𝑡))
35342exbidv 1928 . . . . . 6 (𝑥 = {⟨1o, 1o⟩} → (∃𝑠𝑡 𝑠𝑥𝑡 ↔ ∃𝑠𝑡 𝑠{⟨1o, 1o⟩}𝑡))
3633, 35mpbiri 258 . . . . 5 (𝑥 = {⟨1o, 1o⟩} → ∃𝑠𝑡 𝑠𝑥𝑡)
37 ssid 3967 . . . . . . 7 {1o} ⊆ {1o}
3812rnsnop 6177 . . . . . . 7 ran {⟨1o, 1o⟩} = {1o}
3937, 38, 263sstr4i 3988 . . . . . 6 ran {⟨1o, 1o⟩} ⊆ dom {⟨1o, 1o⟩}
40 rneq 5892 . . . . . . 7 (𝑥 = {⟨1o, 1o⟩} → ran 𝑥 = ran {⟨1o, 1o⟩})
41 dmeq 5860 . . . . . . 7 (𝑥 = {⟨1o, 1o⟩} → dom 𝑥 = dom {⟨1o, 1o⟩})
4240, 41sseq12d 3978 . . . . . 6 (𝑥 = {⟨1o, 1o⟩} → (ran 𝑥 ⊆ dom 𝑥 ↔ ran {⟨1o, 1o⟩} ⊆ dom {⟨1o, 1o⟩}))
4339, 42mpbiri 258 . . . . 5 (𝑥 = {⟨1o, 1o⟩} → ran 𝑥 ⊆ dom 𝑥)
44 pm5.5 362 . . . . 5 ((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → (((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))
4536, 43, 44syl2anc 585 . . . 4 (𝑥 = {⟨1o, 1o⟩} → (((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))
46 breq 5108 . . . . . 6 (𝑥 = {⟨1o, 1o⟩} → ((𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
4746ralbidv 3175 . . . . 5 (𝑥 = {⟨1o, 1o⟩} → (∀𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∀𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
4847exbidv 1925 . . . 4 (𝑥 = {⟨1o, 1o⟩} → (∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∃𝑓𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
4945, 48bitrd 279 . . 3 (𝑥 = {⟨1o, 1o⟩} → (((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
50 ax-dc 10383 . . 3 ((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛))
5122, 49, 50vtocl 3519 . 2 𝑓𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)
5221, 51exlimiiv 1935 1 ω ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  wne 2944  wral 3065  Vcvv 3446  wss 3911  c0 4283  {csn 4587  cop 4593   class class class wbr 5106  dom cdm 5634  ran crn 5635  suc csuc 6320  Fun wfun 6491  cfv 6497  ωcom 7803  1oc1o 8406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673  ax-dc 10383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-fv 6505  df-1o 8413
This theorem is referenced by:  axdc2lem  10385  axdc3lem  10387  axdc4lem  10392  axcclem  10394
  Copyright terms: Public domain W3C validator